Transcript Modelling of recent charge pumping experiments

Modelling of recent
charge pumping experiments
Vyacheslavs (Slava) Kashcheyevs
Mark Buitelaar (Cambridge, UK)
Bernd Kästner (PTB, Germany)
Seminar at
University of Geneva (Switzerland)
April 22st, 2008
Pumping = dc response to
(local) ac perturbation
I
f
weak
adiabatic
(linear in ω)
non-adiabatic
(all orders in ω)
strong
carbon nanotube
carbon nanotube
with acoustic waves with acoustic waves
1st part
GaAs nanowire
with direct gating
Part I: adiabatic pumping in CNT
arXiv:0804.3219
Experimental data
• Peak-and-dip structure
• Correlated with Coulomb blockade peaks
• Reverse wave direction => reverse polarity
Experimental findings
• At small powers of applied acoustic waves the features
grow with power and become more symmetric
• For stronger pumping the maximal current saturates
and opposite sign peaks move aparpt
Experiment
and theory
Interpretation: several dots
Interpretation and a model
(Static) transmission probability
Δ
Two “triple points”
0.3
One “quadruple point”
1
3
Γ/Δ
• If Δ is less than ΓL or ΓR (or both), the two dots
are not resolved in a conductance measurement
Adiabatic pumping (weak + strong)
Charge per period Q
Brouwer /
PTB formula
is easy to obtain
analytically
Q is an integral over
the area enclosed by
the pumping contour
Theory results for pumping
(0,0)
(1,0)
(0,1)
(1,1)
Effects of assymetry
Reduce frequency 5-fold
Conclusions of part I
• Simple single-particle model describes
many experimental features (robust)
• Most detailed experimental test of
the adiabatic pumping theory to-date?
• Alternative mechanisms

Barrier modulation + level renormalization?
 Rectification?
• Work in progress:
 Connect
wih the overlapping peak regime
(moving quantum dot picture, no sign change)
Single-parameter non-adiabatic
quantized charge pumping
weak
adiabatic
(linear in ω)
non-adiabatic
(all orders in ω)
strong
carbon nanotube
carbon nanotube
with acoustic waves with acoustic waves
GaAs nanowire
with direct gating
B. Kaestner, VK, S. Amakawa, L. Li, M. D. Blumenthal, T. J. B. M. Janssen,
G. Hein, K. Pierz, T. Weimann, U. Siegner, and H. W. Schumacher
PRB 77, 153301 (2008) + arXiv:0803.0869
Quantization conditions
weak
adiabatic
(linear in ω)
non-adiabatic
(all orders in ω)
quantization: NO
strong
quantization: YES
with ≥ 2 paramters
quantization: YES
with ≥ 1 paramters
!!! Conflicting mechanisms,
not enough just to tune the frequency
Single-parameter quantization?
• Quantization = loading form
the left + unloading to the right
• One-parameter kills
quantization because
symmetry (i.e. ΓL / ΓR)
at loading and the symmetry
at unloading are the same
 the
• Non-adiabaticity kills
quantization because
 not
enough time to load
and unload a full electron
(0,0)
(1,0)
(1,1)
(0,1)
“Roll-over-the-hill”
Experimental results
V1
V2(mV)
V1
V2
V2
• Fix V1 and V2
• Apply Vac on top of V1
• Measure the current I(V2)
Theory: step 1
• Assume a simple real-space
double-hill potential:
• For every t, solve the “frozen-time”
scattering problem
• Fit the lowest resonance with Breit-Wigner
formula and obtain ε0(t) , ΓL (t) and ΓR (t)
ΓL
ε0
ΓR
Theory: step 2
• Write (an exact) equation-of-motion for P(t)
Flensberg, Pustilnik &
Niu PRB (1999)
• If max(ΓL,ΓR, ω) << kT one gets
a Markovian master equation
For the adiabatic case, see Kashcheyevs, Aharony, Entin,
cond-mat/0308382v1 (section lacking in PRB version)
Results
ε0
• Fix U1(t) and U2
• Solve the scattering
problem for
ε0(t) , ΓL (t) and ΓR (t)
• Fix the frequency and solve the master equation
A: Too slow (almost adiabatic)
Adiabatic limit –
always enough time
to equilibrate,
unloading all we got
from loading to the
same leads
ω<<Γ
Charge re-fluxes back to
where it came from → I ≈ 0
B: Balanced for quantization
Non-adiabatic
blockade of tunneling
allows for left/right
symmetry switch
between loading and
unloading!
ω>>Γ
Loading from the left,
unloading to the right
→I≈ef
C: Too fast
ω
Tunneling is too slow
to catch up with
energy level
switching:
non-adiabaticicty
kills quantization
as expected
The charge is “stuck”
→I≈0
Frequency and gate dependence
I / (ef)
Outlook for part II
• Single-parameter dc pumping possible due
to non-adiabatic blockade of tunneling
• In progress:
“bare-bones” model
for quantitative fitting
 same type of pump with carbon nanotubes
 two-parameter
• In the same device there exists a range of
qunatized ac current! not measured (yet)